Parametric and internal resonances of in-plane accelerating viscoelastic plates

This paper analytically and numerically investigates the nonlinear vibration in parametric and internal resonances of in-plane accelerating viscoelastic plates subjected to plane stresses. An approximate nonlinear plate theory was developed under the Kirchoff assumptions. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean axial speed. The governing equation with the associated boundary conditions is derived from the generalized Hamilton principle and the Kelvin constitutive relation. The method of multiple scales is applied to establish the solvability conditions in principal parametric and internal resonances. The steady-state responses are predicted in three possible patterns: trivial, single-mode, and two-mode solutions. The stabilities of the steady-state responses are determined based on the Routh-Hurwitz criterion. The effects of the mean in-plane translating speed, the in-plane translating speed fluctuation amplitude, the viscosity coefficient, and the nonlinear coefficient on the steady-state responses are examined. The differential quadrature schemes are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the trivial and single-mode solutions of the steady-state responses.